Make sure you read to my last comment for the most important thing
about this project.

OK - here is what you could do. Just make a platform for cushions. Make
it L shaped to sit in a corner. if you can fasten it to the wall you
will have that much more support and could do w/o learning about
torsion boxes.

Assuming an L shape

You will need:
Materials
two sheets of 3/5x4x8 plywood or MDF - 2x$45 = $90 (Good stuff, A/B or
a hardwood veneer if possible, you will paint it)
2x4s - qty 6 @ $2.69 = $16.14 Use these to form up the frame that the
plywood will attach to.
Screws - One box of 1 5/8" multi purpose screws (Drywall screws) and
one box of 3" deck screws to hold the frame together. about $6
sandpaper $3
paint = $20
glue = $4
Cushions = Who knows, I suspect you could end up spending a lot here
Materials, without cushions = $139

Tools:
Saw to cut the plywood (Borrow one from a neighbor) = Free
Drill/driver = (Again Borrow one) = Free

OK, so even if you spent nothing on tools I cant really imagine
building a decent base for under $150. A crude base is possible at that
price. There could be some techniques for making this not too bad with
the above materials but would require more skills and tools.

Post here if you want some suggestions for making the base sort of
niceish with the above materials.

Here is a suggestion - try and make just a coffee table in the style if
you posted picture without any cushions. If you can do a decent job of
that then move on from there. If you are dead set on a couch then I
would suggest starting with the cushions. Price them out. If you find
the right cushions then you can design your platform around them.

Oh - And one more thing. Once you get established, have more money,
find a wife and move to the burbs. THERE IS NO WAY IN HELL YOU WIFE
WILL LET YOU MOVE THAT THING INTO YOUR HOUSE.

FYI, I have a coffee table I built in college. Modeled after the tables
at chili's restaurants. Tile top, etc. It came into the house for a
short time but was banished pretty quickly.